Construct a new solution to Exercise B.T50 along the following outline

Construct a new solution to Exercise B.T50 along the following outline.From the n n matrix A, construct the linear transformation T : Cn→Cn, T (x) = Ax. Use Theorem NI, Theorem IMILT and Theorem ILTIS to translate between the nonsingularity of A and the surjectivity/injectivity of T . Then apply Theorem ILTB and Theorem SLTB to connect these properties with bases.